Homework Help: Use integration to find the sum of a series

1. May 8, 2013

raopeng

1. The problem statement, all variables and given/known data
Find the sum using integration: $lim_{n→∞} \frac{n}{(n+1)^2} + ... + \frac{n}{(2n)^2}$

2. Relevant equations

3. The attempt at a solution
I think this requires a clever construction of a series of an finite integral which after integration gives the series. Then it can be solved by summing up the series inside the integral then integrate the whole thing. But now what is bothering me is how to construct such a function, every function I have tried seems remotely far from the series in the question. Thanks for your time.

2. May 8, 2013

haruspex

How about bounding the sum by a pair of integrals over (n, 2n)?

3. May 8, 2013

Dick

Or show that the given sum is equivalent to a Riemann sum for some function. So taking the limit is equivalent to finding the integral.

4. May 9, 2013

raopeng

Oh thanks so much!

5. May 9, 2013

raopeng

I transform the series in this way: $Ʃ_{k=1} ^{n} \frac{1}{(1+k/n)^2 n}$ which turns into an integral $\int^1 _0 \frac{dx}{(1+x)^2}$. An integration gives 1/2.